Cycle indicator polynomials

We have seen in Pólya's Theorem that the generating
function for the enumeration of
the G-classes on YX by weight is equal to

(1)/( | G | )ågÎGÕi=1 | X | (
åyÎYyi)ai(bar (g)).

This polynomial can be obtained from the polynomial

C(G,X):=(1)/( | G | )ågÎGÕi=1 | X
| ziai(bar (g))ÎQ[z1,...,z | X | ]

by simply replacing the indeterminate zi by the polynomial åyyi.
It is therefore the polynomial C(G,X) which really matters. We call this
polynomial the cycle indicator polynomial
or the cycle index ofGX, since it
displays the cycle structure of the elements bar (g)Îbar (G). (But
it should be mentioned that C(G,X)=C(H,X) does not mean that
bar (G) and bar (H) are isomorphic. There are counterexamples known:
for example, the regular representation of the nonabelian group of order
p3, p an odd prime number, has the same cycle indicator polynomial as the
regular representation of the abelian group Cp´Cp´Cp
(exercise.)
In the case when we wish to display its indeterminates we shall write
C(G,X;z1,...,z | X | ), and if we replace zi by riÎQ, we simply write C(G,X;r1,...,r | X | ). Hence we
obtain, for
example (cf. Theorem):

and call this process Pólya-substitution.
Using this notation we
can rephrase corollary as follows:

Pólya's Theorem
The generating
function for the
enumeration of G-classes on YX by content can be obtained from the
cycle indicator of GX by Pólya-substituting åyÎYy into
the cycle indicator polynomial C(G,X). Hence this generating function
is equal to

C(G,X | åyÎYy).

In order to count G-classes by content it therefore remains to evaluate
the cycle indicator of GX. A few examples should be welcome:

Examples: of cycle indicator polynomials

The cycle indicator of the identity subgroup
of Sn and its natural
action on n is

The cycle indicator of the natural
action of the dihedral group Dn
of order 2n on the set n (which can be considered as being the
set of vertices of the regular n-gon, of which Dn is the
symmetry group) satisfies

C(Dn,n)=(1)/(2)C(Cn,n)+(1)/(2)z1z2(n-1)/2
if n is odd,

C(Dn,n)=(1)/(2)C(Cn,n)+(1)/(4)(z2n/2+z12z2(n-2)/2)
if n is even.

This follows from the fact that the rotations form a cyclic subgroup of order
n, while the remaining reflections either leave exactly one vertex fixed
(in the case when n is odd) or two vertices or none (in the case when n is
even), and group the remaining vertices into pairs of vertices that are
mapped onto each other.

In order to make life easier we can replace
y0+...+ym-1 by
1+y1+...+ym-1, and in the case when | Y | =2 we can even
take 1+y instead of y0+y1 or 1+y1,so that, for example,
the generating function for the graphs on 4 vertices by their number of edges
takes the following form:

while the number of selfcomplementary graphs on v vertices is equal to

C(Sv,[v choose 2];0,2,0,2,...).

The number of graphs on v vertices which contain e edges is the
coefficient of ye in the polynomial

C(Sv,[v choose 2] | 1+y).

These examples have shown that the evaluation of the cycle indicator polynomial
is an important step towards the solution of various enumeration problems, so
that a few remarks concerning this are well in order. The cycle indicators of
the natural actions of Sn, An, Cn, and Dn
are at hand and we therefore put the question how we can obtain
cycle indicators of groups, which are direct sums or certain
products of symmetric, alternating, cyclic or dihedral groups. More generally
we want to know how the cycle indicator of a sum or a product can be
obtained from the cycle indicators of the summands or factors, depending of
course on the action in question. The following first remarks are clear
from the definitions
of direct sum and cartesian product:

Further important constructions are the plethysmH G, and the composition.
They are defined as permutation
groups induced by similar actions of H wr X G , and so the
corresponding cycle indicator polynomials are the same:

Finally we show how the cycle indicator polynomial C(G,X)
can be obtained from the character
c:g -> | Xg |
of the action GX in question. In order to do this we have to evaluate
the numbers ai(bar (g)),iÎN, for each gÎG,
and we shall do this
by inverting the equations

c(gk)=a1(bar (gk))=a1(bar (g)k)=åd | kd·ad(bar (g))

(cf. the lemma describing the powers of a cycle
for the last equation). The inversion procedure uses the number
theoretic Moebius function and the corresponding inversion theorem, but as
we are going to use also other Moebius functions later on we shall describe
this inversion method in a slightly
more general context than is actually needed
here.

We start doing so by introducing the notion of incidence algebra.
Let
(P, <= ) denote a poset
(partially ordered set). Its partial order <=
allows us to introduce intervals[p,q]:={rÎP | p <= r <= q}. If
all these intervals are finite, then (P, <= ) is called a locally finiteposet.
Assuming this and denoting by F a field, we can turn the set

IF(P):={j:P2 -> F | j(p,q)=0
unless
p <= q}

of all the incidence functions
into an F-algebra. The following
addition and scalar multiplication define a vector space structure:

This makes IF(P) into a ring (exercise). The
identity element is the Kronecker d-function

d(p,q):=1 if p=q

d(p,q):=0 otherwise.

As scalar mutiplication and convolution satisfy

r(jy)=(rj)y=j (ry),

this ring is even an F-algebra, the incidence
algebra
overF of (P, <= ). It is important to characterize its
invertible elements, i.e. the j in IF(P) for which there
exists a yÎIF(P) such that yj=d and
also jy=d. We obtain these incidence functions
by an easy argument from linear algebra that allows to identify the
incidence functions with upper triangular matrices as follows.
In the case when P is finite, we can embed the
partial order into a total order (i.e. we can
number the pÎP in a way such that
pi<pk implies i<k). This yields a bijection between IF(P) and
the set of upper triangular matrices over F which respects addition and
scalar multiplication. The convolution product corresponds to the
matrix product of the associated matrices

j -> F:=(j(pi,pk)),

and so j is invertible if and only if the values j(p,p)
are nonzero. But this is true also in the more general case when we only
assume (P,£) to be locally finite, since then we can easily show
that the incidence function recursively defined
in the second item of the following
lemma is in fact an inverse with respect to the convolution product:

Lemma:
For each locally finite partial order (P,£)
and any jÎIF(P)
the following is true:

j is invertible if and only if, for
each pÎP, we have

j(p,p)
not = 0.

If j is invertible, then j-1 satisfies
j-1(p,p)=j(p,p)-1, and

j-1(p,q)
=-j(q,q)-1årÎ[p,q)j-1(p,r)j(r,q)

=-j(p,p)-1årÎ(p,q]j(p,r)j-1(r,q),

where as usual [p,q) and (p,q] denote half open intervals.

A specific invertible incidence function is the zeta functionwhich describes the partial order in question:

z(p,q):=1 if p <= q,

z(p,q):=0 otherwise.

Its inverse is called the Moebius function of(P, <= ):
m:=z-1,
for which we obtain from the preceeding lemma the following recursions:

m(p,q)=-årÎ[p,q)m(p,r)=-årÎ(p,q]m(r,q).

This close connection between the zeta and the Moebius function
provides a very useful inversion theorem which we
introduce next. In a poset (P, <= ) the sets {qÎP | q <= p}
are called the principal (order-)ideals
ofP, while the sets
{qÎP | q >= p} are called the principal filters
. The
inversion theorem now reads as follows:

The Moebius Inversion
Let(P, <= ) denote a
locally finite
poset and let F and G denote mappings from P into the field F. Then

if all the principal ideals of P are finite, we have the following
equivalence of systems of equations:

" p: G(p)=åq <= pF(q)
iff " p: F(p)=åq <= pG(q)m(q,p).

If all the principal filters of P are finite, we have
the following equivalence of systems of equations:

" p: G(p)=åq >= pF(q)
iff " p: F(p)=åq >= pm(p,q)G(q).

Proof: We prove the first statement, the second one follows analogously.
We add to P an element 0 such that 0<p, for
each pÎP, obtaining
a new poset (bar (P)=PÈ{0}, <= ). As all the principal ideals of
(P, <= ) are supposed to be finite, (bar (P), <= ) is a locally finite
poset. Now we associate to F and G the incidence functions jyÎIF(bar (P)) defined by

j(p,q):=F(q) if p=0 and q>0,

j(p,q):=0 otherwise,

y(p,q):=G(q) if p=0 and q>0,

y(p,q):=0 otherwise.

The system of equations G(p)=åq <= pF(q) is equivalent to the identity
y=jz, which again is equivalent to ym=j,
and this is nothing but the system of equations F(p)=åq <= p
G(q)m(q,p).

A particular locally finite poset is (N*, | ), the set
of positive natural numbers
together with divisibility as its partial order. The corresponding m
is called the number theoretic
Moebius function. Instead of
m(p,q) one can write m(q/p) in this case since
the intervals [p,q] and [r,s] are order isomorphic if q/p=s/r, and so
m(p,q)=m(r,s), by the above mentioned recursion.
Thus we obtain, by an application of the Moebius Inversion
formula and the formula for the character
of gk:

Corollary:
For each finite
action GX the following equivalent
systems of equations hold:

where m denotes the number theoretic Moebius function. In particular we
obtain the following expression of the cycle indicator of GX in terms of
the character c of GX:

C(G,X)=(1)/( | G | )ågÕizii-1Sd | im(i/d)c(gd).

In order to apply this result, we need to know the values
of the number theoretic Moebius function. As z is defined by the partial
order, the same holds for the Moebius function, and hence the Moebius function
is the same for order isomorphic posets. Thus the values of the Moebius
function on (N*, | ) can be evaluated by noting that the
Moebius function is multiplicative and that an
interval [d,n], where d divides n, is order isomorphic to the
cartesian product

[1,p1k1]´...´[1,prkr],

if n/d has the prime number decomposition n/d=p1k1·...·prkr, where the pi are different prime numbers and the ki >= 1.
Thus, for the number theoretic Moebius function we have
(exercise):